To determine the rate of change from the table provided, we need to look at how the values change in relation to the input values.
From the data:
- When \( x = 2 \), \( y = 19 \)
- When \( x = 3 \), \( y = 26 \)
- When \( x = 4 \), \( y = 33 \)
- When \( x = 5 \), \( y = 40 \)
- When \( x = 6 \), \( y = 47 \)
To find the rate of change, calculate the change in \( y \) divided by the change in \( x \).
For example:
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From \( x = 2 \) to \( x = 3 \): \[ \Delta y = 26 - 19 = 7, \quad \Delta x = 3 - 2 = 1 \quad \Rightarrow \quad \text{Rate of change} = \frac{\Delta y}{\Delta x} = \frac{7}{1} = 7 \]
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From \( x = 3 \) to \( x = 4 \): \[ \Delta y = 33 - 26 = 7, \quad \Delta x = 4 - 3 = 1 \quad \Rightarrow \quad \text{Rate of change} = \frac{7}{1} = 7 \]
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From \( x = 4 \) to \( x = 5 \): \[ \Delta y = 40 - 33 = 7, \quad \Delta x = 5 - 4 = 1 \quad \Rightarrow \quad \text{Rate of change} = \frac{7}{1} = 7 \]
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From \( x = 5 \) to \( x = 6 \): \[ \Delta y = 47 - 40 = 7, \quad \Delta x = 6 - 5 = 1 \quad \Rightarrow \quad \text{Rate of change} = \frac{7}{1} = 7 \]
The rate of change is consistent across all intervals and is equal to 7.
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